Given the potential of low-earth orbit (LEO) satellites in terms of navigation enhancement, accurately estimating the differential code bias (DCB) of GNSS satellites and LEO satellites is an important research topic. In this study, to obtain accurate DCB estimates, the effects of vertical total electron content (VTEC) modeling parameters of the topside ionosphere on DCB estimation were investigated using LEO observations for the first time. Different modeling parameters were set in the DCB estimations, encompassing modeling spacing in the dynamic temporal mode and degree and order (D&O) in spherical harmonic modeling. The DCB precisions were then evaluated, and the impacts were analyzed. Thus, a number of crucial and beneficial conclusions are drawn: (
Keywords: GNSS; GPS; LEO; GPS satellite differential code biases (DCB); LEO receiver DCB; LEO-based vertical total electron content (VTEC); modeling spacing; modeling degree and order (D&O)
Differential code bias (DCB) is a critical error source in navigation positioning and ionosphere modeling [[
With the launch of tens of thousands of LEO satellites in the future, accurately estimating the DCB of GNSS and LEO satellites is an important research topic. In recent years, scholars have studied and used LEO onboard observation data to estimate DCB parameters. A few scholars estimated the DCBs of the GPS satellite and LEO receiver as unknown parameters simultaneously [[
In this paper, the impacts of topside ionosphere VTEC modeling parameters on LEO-based DCB estimation are investigated and analyzed for the first time to obtain superior modeling parameters and accurate DCB estimates. The research conclusions can provide a reference for LEO-augmented DCB estimation. Section 2 presents the model and strategy for GPS and LEO DCB estimation using LEO observation data. Section 3 analyzes the results of experiments in which different modeling parameters are set in the DCB estimations, encompassing the modeling spacing in the dynamic temporal mode and the degree and order in spherical harmonic modeling, employing GRACE-FO [[
The geometry-free (GF) combination observation estimation method involves a simpler calculation and does not require outlier information. Therefore, we employed GF combinations of pseudo-range observations to estimate the DCB parameters. The DCB values (P1-P2) for the GPS satellites and LEO receivers were estimated daily as constant values and simultaneously with the topside ionosphere VTEC parameters.
DCB estimation using LEO observations is not affected by the troposphere. Hence, dual-frequency code observations are commonly expressed as shown in Equation (
(
(
Here scripts s and r denote the GPS satellite and LEO receiver; 1 and 2 denote frequency numbers;
In this study, we screen code observation data using residuals and LEO orbits derived from phase observations, and the preprocessing for pseudo-range also draws on the onboard data preprocessing method of LEO precise orbit determination. The pseudo-range is relatively clean in this way.
The F&K mapping function [[
(
where
Introduced Equations (
(
Here the vector Z with a total of n rows is related to DCB and VTEC modeling coefficients; F is the design matrix that consists of the matrix A, B, and C, related to GNSS DCB, LEO DCB, and VTEC modeling coefficients; and
Considering that the DCBs of the GPS satellite and LEO receiver are closely correlated, the DCB datum is defined by a zero-mean condition in the DCB estimation for de-correlation. Equation (
(
Here
Daily DCB values were realigned by applying a shift from non-all satellite DCB values, which were calculated using a common set of satellites over the study period. After alignment [[
(
Here scripts s and d denote one satellite and one day, respectively; D refers to the total days of a month;
The RMS values for GPS DCB estimates with respect to external reference products after conducting the alignment procedures is calculated as follows:
(
Here
To obtain suitable and superior topside ionospheric VTEC modeling parameters in LEO-based DCB estimation, we investigated the effects of the different modeling parameters on DCB estimates and precision, including modeling spacings in dynamic temporal mode and degrees and orders in spherical harmonic modeling. The precision of DCB estimates was then evaluated, and the effects of the parameters were analyzed. We used onboard GPS observation data from the twin GRACE-FO satellites (GRCC and GRCD) from 1–30 January 2019 to estimate the GPS and LEO satellite DCBs. We comprehensively evaluated the internal and external consistency precisions of GPS DCB estimation and compared the monthly stability of GPS and LEO DCB estimates as internal precision, which was presented STD. The comparisons between the GPS DCB estimates and CODE DCB products represent the external precision in the form of the mean and RMS of the difference between the estimates and CODE products. The LEO DCB can provide internal precision but not external precision owing to a lack of external reference products.
Additionally, this study requires the precise orbits of the twin GRACE-FO satellites owing to the needs of Equation (
To investigate the effects of modeling spacings on DCB estimate and precision and to obtain superior parameter settings, we set different modeling spacings of 2H, 4H, 6H, and 12H in the LEO-based VTEC dynamic temporal modeling mode. The modeling degree and order were set to 8. Then, the effects of modeling spacing on DCB estimation were analyzed by evaluating the precision of the GPS and LEO DCB estimates.
The mean DCB can exhibit the averaged ranges of DCBs in the form of specific and intuitive numbers; thereby, Table 1 presents the mean values of GPS satellite DCB estimates after adopting different modeling spacings of 2H, 4H, 6H, and 12H using the GRCC (a) and GRCD (b) observations. The GPS DCB values for G04 were not estimated owing to a lack of observations. The mean GPS DCB estimates range between −10 and 10 ns and are stable. There is no significant difference in mean GPS DCBs with different modeling spacings. Considering the limitation of mean statistics in Table 1 in terms of reflecting all DCB values, Figure 1 displays the maximum differences in the GPS satellite DCB estimates after adopting different modeling spacings using GRCC and GRCD observations during the experiment period. The differences in the GPS DCB estimates using different modeling spacings are in the range of 0.05 ns. Specifically, in Figure 1, the difference in G15 DCB estimates adopting different modeling spacings is the maximum value, 0.04 ns, when using GRCC data, whereas for GRCD data, the difference in G12 DCB is the largest, approximately 0.05 ns. Therefore, corresponding to Figure 1, the maximum, minimum, and differences in G15 and G12 DCB estimated based on different modeling spacings using GRCC and GRCD observations, respectively, are presented in Table 2. In Table 2a using GRCC data, the difference in the G15 DCB estimates based on different modeling spacings on DOY 5 is the largest, and the maximum, minimum, and difference are 2.6815, 2.6415, and 0.0400 ns, respectively. In Table 2b using GRCD data, the difference in G12 DCB estimated on DOY 4 is the largest, and the maximum, minimum, and difference are 4.3340, 4.2874, and 0.0466 ns, respectively. Therefore, there is a difference in the GPS DCB estimated using different modeling spacings, within 0.05 ns. The different modeling spacings affect the GPS DCB estimates.
Figure 2 presents the monthly stabilities (STD) of the GPS satellite DCB estimates adopting different modeling spacings of 2H, 4H, 6H, and 12H using GRCC (top) and GRCD (bottom) observation data. The STDs of the GPS DCB estimates using GRCC and GRCD data are in the range of 0.11 ns with high stability. Meanwhile, the maximum differences in the STD of the GPS DCB estimates adopting different modeling spacings are shown in Figure 3. The maximum differences in the STD of the GPS DCB using different modeling spacings are within 0.006 ns. The STD values of the GPS DCB estimates adopting different modeling spacings exhibit no significant differences.
Table 3 presents the mean STDs of the GPS DCB estimates adopting the different modeling spacings. Statistically, the mean STD values of GPS DCBs with modeling spacings of 2H, 4H, 6H, and 12H using GRCC and GRCD data are 0.0624, 0.0621, 0.0620, and 0.0619 ns and 0.0648, 0.0643, 0.0641, and 0.0639 ns, respectively. The GPS DCB estimates with modeling spacings of 12H have the optimal mean STD results. The GPS DCB estimates using twin LEO satellite data attain similar and high stability.
Figure 4 depicts the mean differences for GPS DCB estimates with different modeling spacings relative to CODE products. The mean differences in Figure 4 vary between −0.6 and 0.8 ns, which indicates that the GPS DCBs estimated by LEO satellite and ground station data have good consistency. Figure 5 showcases the RMS of the differences between the GPS DCB estimates adopting different modeling spacings (2H, 4H, 6H, and 12H) and the CODE DCB products. To visually demonstrate the differences in RMS of GPS DCBs with different modeling spacings, Figure 6 displays the maximum differences in RMS of GPS DCB estimates using different modeling spacings. The maximum differences in RMS of GPS DCBs using different modeling spacings are in the range of 0.03 ns. The different modeling spacings affect the RMS of the GPS DCB estimates. The corresponding mean RMS values are listed in Table 4. The RMS results of the GPS DCBs with different modeling spacings relative to the CODE products are in the range of 0.8 ns. The mean RMSs of the GPS DCBs are in the range of 0.3 ns, which indicates that the GPS DCBs estimated by LEO satellite and ground station data have good consistency. Statistically, the mean RMS results in Table 4 indicate that the GPS DCBs with a modeling spacing of 12H had slightly smaller RMS values than the others. Additionally, the GPS DCB estimates using twin satellite data have similar precisions, and the RMS results using GRCC observations are slightly poorer than those obtained using GRCD data.
Figure 7 presents the time series of the receiver DCB estimates with modeling spacings of 2H, 4H, 6H, and 12H for the twin GRACE-FO satellites. The GRCC receiver DCB estimates are located at approximately −1.0 ns, whereas the GRCD DCB estimates fluctuate at approximately 4.6 ns, and their receiver DCB estimates are stable. In Figure 7, the GRCC and GRCD receiver DCB estimates are sorted in descending order as follows: receiver DCBs with modeling spacings of 12H, 6H, 4H, and 2H, where the receiver DCBs decrease as the modeling spacing decreases.
In order to visually demonstrate the differences in receiver DCBs with different modeling spacings, Figure 8 exhibits the maximum differences in the LEO receiver DCB estimates using different modeling spacings. Figure 9 shows the maximum differences in the STD of the receiver DCB using different modeling spacings. The maximum difference in receiver DCBs using different modeling spacings is 0.22 ns, whereas the maximum differences in STD of receiver DCBs are within 0.005 ns. This indicates that different modeling spacings have a significant impact on receiver DCB estimates.
The mean values and STD results for the LEO receiver DCB estimates with different modeling spacings are listed in Table 5. Statistically, the maximum receiver DCB values for both the GRCC and GRCD are the DCBs with modeling spacings of 12H, whereas the minimum receiver DCBs are those with modeling spacings of 2H. The differences in the mean receiver DCBs with different modeling spacings for the GRCC and GRCD are in the ranges of 0.1569 and 0.1684 ns, respectively. The GRCC and GRCD receiver DCBs achieve optimal STD results when modeling spacings of 6H and 4H are applied. The receiver DCBs of the twin GRACE-FO satellites are different; however, their DCB estimates have similar STD results.
In summary, the GPS DCB estimates using a modeling spacing of 12H have higher precision than the others, whereas LEO receiver DCBs applying the modeling spacings of 4H or 6H obtain optimal STD.
This section investigates the impact of the D&O of spherical harmonic modeling on DCB estimates and precision, thereby obtaining suitable modeling D&O parameters. The 6, 8, and 10D&Os of spherical harmonic modeling were introduced into topside ionosphere modeling and DCB estimation. Subsequently, the GPS and LEO DCB estimates were analyzed and evaluated. The modeling spacing was fixed at 4H to improve the temporal resolution.
The mean values of GPS DCB estimates adopting different modeling parameters of 6, 8, and 10D&Os using the GRCC (a) and GRCD (b) observations are listed in Table 6. The GPS DCB values for G04 were not estimated owing to a lack of observations. The mean GPS DCB estimates range between −10 and 10 ns and are stable. There is no significant difference in the GPS DCB estimates obtained using different modeling D&Os. Considering the limitation of mean statistics in Table 6, Figure 10 exhibits the maximum differences in GPS DCB estimates adopting different modeling D&Os using GRCC and GRCD data. The differences in the GPS DCB estimates using different modeling D&Os are in the range of 0.05 ns. It indicates that the effects of the modeling spacing and D&O on GPS DCB estimates are at the same level. Specifically, the differences in G05 DCB estimates adopting different modeling D&Os using GRCC and GRCD data are both the largest, within 0.04 and 0.05 ns, respectively. The maximum, minimum, and differences in G05 DCB estimated with different modeling D&Os using GRCC and GRCD observations are presented in Table 7. In Table 7a, using GRCC data, the difference in G05 DCB estimates between different modeling D&Os on DOY 13 is the largest, and the maximum, minimum, and difference are 3.0099, 2.9776, and 0.0323 ns, respectively. In Table 7b, using GRCD data, the difference in G12 DCB estimated on DOY 4 is the largest, and the maximum, minimum, and difference are 3.0515, 3.0048, and 0.0467 ns, respectively. Therefore, there is a certain difference in the GPS DCB estimates using different modeling D&Os, all within 0.05 ns. The different modeling D&Os have certain effects on the GPS DCB estimates.
Figure 11 showcases the STD of GPS DCB estimates adopting different modeling D&Os of 6, 8, and 10 using GRCC and GRCD data. The STDs of GPS DCB estimates based on GRCC and GRCD data are within 0.11 ns. Additionally, Figure 12 displays the maximum differences in the STD of the GPS DCB estimates by applying different modeling D&Os. The maximum differences are within 0.005 ns. The STDs of the GPS DCB estimates adopting different modeling D&Os show no marked differences.
Table 8 presents the mean STDs of GPS satellite DCB estimates by applying different modeling D&Os. Statistically, the mean STD values of GPS DCBs with modeling D&Os of 6, 8, and 10 using GRCC and GRCD data are 0.0620, 0.0621, and 0.0623 ns and 0.0643, 0.0643, and 0.0644 ns, respectively. The GPS DCB estimates, with modeling D&Os of 6 and 8, attain the optimal STD results.
Figure 13 exhibits the mean differences for GPS DCB estimates with different D&Os relative to CODE products. The mean differences in Figure 13 vary between −0.6 and 0.7 ns, which indicates that the GPS DCBs estimated by LEO satellite and ground station data have good consistency. Figure 14 presents the RMS of the differences between the GPS DCB estimates adopting different modeling D&Os of 6, 8, and 10 and the CODE DCB products. To visually exhibit the differences in the RMS, Figure 15 displays the maximum differences in the RMS of the GPS DCB estimates using different modeling D&Os. The maximum differences reach 0.02 ns. The different modeling D&Os have certain effects on the RMS of the GPS DCB estimates. The corresponding mean RMS values are listed in Table 9. These RMS results are in the range of 0.8 ns. The mean RMS values are in the range of 0.3 ns, which indicates that the GPS DCBs estimated by LEO satellite and ground station data have good consistency. Statistically, Table 9 shows that the GPS DCBs with modeling D&Os of 8 and 10 have slightly higher RMS than the others. The RMS results using the GRCD observations are slightly higher than those obtained using the GRCC data. Considering the STD and RMS results of the GPS DCBs, the GPS DCB estimates with modeling D&Os of 8 attain superior precision.
Figure 16 exhibits the time series of the receiver DCB estimates with different modeling D&Os of 6, 8, and 10 for twin GRACE-FO satellites. The GRCC receiver DCB estimates are located at approximately −1.0 ns, whereas the GRCD DCB estimates fluctuate at approximately 4.55 ns. The GRCC and GRCD receiver DCBs with the 6 D&O in spherical harmonic modeling are greater than the others. In order to visually present the differences in receiver DCBs with different modeling D&Os, Figure 17 shows the maximum differences in LEO DCB estimates using different modeling D&Os. Meanwhile, Figure 18 displays the maximum differences in the STD of the receiver DCB using different modeling D&Os. The maximum differences are within 0.02 ns, inferior to the differences in the GPS DCBs, while the maximum differences in STDs are within 0.002 ns.
The mean values and STD results for the receiver DCB estimates with different modeling D&Os are listed in Table 10. Statistically, the mean receiver DCBs for GRCC and GRCD are both sorted in descending order as follows: receiver DCBs with 6, 8, and 10D&Os modeling, where the receiver DCBs decrease as the modeling D&Os increase. The GRCC receiver DCBs with 10D&O modeling gain the minimum STD result, whereas the GRCD DCBs applying 8 and 10D&Os have a better STD than the others. The receiver DCBs of the GRCC and GRCD differ; however, their DCB estimates have similar STD results.
In summary, adopting 8D&O in the LEO-based VTEC modeling exhibits superior estimates and precisions for both GPS and LEO DCBs. Additionally, compared to the results in Section 3.1, this demonstrates that the impact of modeling spacing on DCB estimates is greater than that of modeling D&O.
The effects of topside ionosphere VTEC modeling parameters on LEO-based DCB estimates and precision were investigated using GRACE-FO observation data to obtain superior modeling parameters. Different modeling parameters were set into the DCB estimations, encompassing the modeling spacing in the dynamic temporal mode and D&O in LEO-based VTEC modeling. The differences in GPS and LEO DCB estimates with the different modeling parameters were showcased, the precision of the DCB estimates was evaluated, and the effects of these parameters on DCB estimation were analyzed. The beneficial conclusions are drawn as follows:
(
The maximum difference in receiver DCBs adopting different modeling spacings is 0.22 ns, which indicates that modeling spacing has a significant impact on the receiver DCBs compared with GPS DCBs. The GRCC and GRCD receiver DCBs gain more optimal STD than the others when applying modeling spacings of 6H and 4H, respectively.
In summary, the GPS DCB estimates using a modeling spacing of 12H have higher precisions than the others, whereas LEO DCBs applying the modeling spacings of 4H or 6H obtain optimal STD results.
(
In terms of receiver DCBs, the maximum differences in receiver DCBs using different modeling D&Os are within 0.02 ns, inferior to the differences in the GPS DCBs. The GRCC and GRCD DCBs achieve the optimal STD when adopting modeling D&Os of 10 and 8, respectively.
In summary, adopting 8D&O in the LEO-based VTEC modeling can obtain superior estimates and precisions for both GPS and LEO DCBs.
(
Graph: Figure 1 Maximum differences in GPS DCB estimates adopting different modeling spacings using twin LEO observations.
Graph: Figure 2 Monthly stability (STD) of GPS DCB estimates adopting different modeling spacings ((top): GRCC and (below): GRCD data).
Graph: Figure 3 Maximum differences in STD of GPS DCB estimates adopting different modeling spacings.
Graph: Figure 4 Mean differences for GPS DCBs using different spacings relative to CODE products.
Graph: remotesensing-15-05335-g004b.tif
Graph: Figure 5 Root-mean square (RMS) for differences between the GPS DCB estimates using different modeling spacings and CODE DCB products ((top): GRCC and (below): GRCD data).
Graph: Figure 6 Maximum differences in RMS of the GPS DCB estimates using different modeling spacings.
Graph: Figure 7 GRACE-FO receiver DCB estimates with different modeling spacings ((top): GRCC and (below): GRCD).
Graph: Figure 8 Maximum differences in LEO DCB estimates using different modeling spacings.
Graph: Figure 9 Maximum differences in STD of LEO DCB estimates using different spacings.
Graph: Figure 10 Maximum differences in GPS DCB estimates adopting different D&Os using twin LEO observations.
Graph: Figure 11 Monthly stability of GPS satellite DCB estimates using different modeling D&Os ((top): GRCC obs. and (below): GRCD obs.).
Graph: Figure 12 Maximum differences in STD of GPS DCB estimates using different D&Os.
Graph: Figure 13 Mean differences for GPS DCBs using different D&Os relative to CODE products.
Graph: Figure 14 RMS of differences between the GPS DCB estimates using different modeling D&Os and CODE DCB products ((top): GRCC data and (below): GRCD data).
Graph: Figure 15 Maximum differences in RMS of the GPS DCB estimates using different D&Os.
Graph: Figure 16 GRACE-FO receiver DCB estimates adopting different modeling D&Os ((top): GRCC and (below): GRCD).
Graph: Figure 17 Maximum differences in LEO DCB estimates using different modeling D&Os.
Graph: Figure 18 Maximum differences in STD of LEO DCB estimates using different D&Os.
Table 1 Mean GPS differential code bias (DCB) estimates based on different modeling spacings using GRCC (a) and GRCD (b) observations [ns].
G01 −7.4145 −7.4174 −7.4183 −7.4191 G02 9.1243 9.1298 9.1330 9.1375 G03 −5.0039 −5.0161 −5.0208 −5.0262 G05 2.9826 2.9782 2.9775 2.9772 G06 −6.6772 −6.6762 −6.6750 −6.6726 G07 3.3569 3.3536 3.3532 3.3534 G08 −7.5298 −7.5226 −7.5205 −7.5182 G09 −4.9578 −4.9469 −4.9431 −4.9388 G10 −5.0343 −5.0449 −5.0487 −5.0529 G11 3.7213 3.7332 3.7374 3.7420 G12 4.1600 4.1480 4.1436 4.1385 G13 3.4175 3.4277 3.4307 3.4336 G14 1.9124 1.9205 1.9223 1.9231 G15 2.6321 2.6467 2.6523 2.6588 G16 2.7966 2.7844 2.7801 2.7753 G17 2.9321 2.9406 2.9429 2.9452 G18 −0.2903 −0.2915 −0.2913 −0.2908 G19 5.6329 5.6405 5.6425 5.6440 G20 1.4707 1.4640 1.4621 1.4597 G21 2.2743 2.2791 2.2821 2.2857 G22 7.2749 7.2648 7.2612 7.2572 G23 9.1329 9.1434 9.1469 9.1511 G24 −5.5552 −5.5542 −5.5536 −5.5526 G25 −7.1077 −7.1219 −7.1273 −7.1336 G26 −8.5698 −8.5834 −8.5883 −8.5941 G27 −5.2432 −5.2366 −5.2348 −5.2330 G28 3.2078 3.1953 3.1910 3.1861 G29 2.2061 2.2135 2.2154 2.2167 G30 −6.0024 −6.0065 −6.0074 −6.0079 G31 4.9635 4.9588 4.9573 4.9560 G32 −4.3827 −4.3735 −4.3709 −4.3688 G01 −7.2909 −7.2886 −7.2910 −7.2904 G02 9.3020 9.3094 9.3125 9.3178 G03 −4.8126 −4.8174 −4.8270 −4.8316 G05 3.0288 3.0263 3.0288 3.0305 G06 −6.6031 −6.6008 −6.5987 −6.5960 G07 3.4199 3.4194 3.4154 3.4150 G08 −7.2286 −7.2240 −7.2212 −7.2195 G09 −4.7793 −4.7650 −4.7637 −4.7593 G10 −4.9828 −4.9938 −4.9933 −4.9963 G11 3.7889 3.8041 3.8055 3.8099 G12 4.2406 4.2284 4.2184 4.2111 G13 3.4716 3.4804 3.4854 3.4884 G14 2.0112 2.0114 2.0217 2.0232 G15 2.6436 2.6580 2.6637 2.6704 G16 2.9451 2.9338 2.9275 2.9224 G17 2.9843 2.9941 2.9956 2.9973 G18 −0.2142 −0.2115 −0.2117 −0.2098 G19 5.7302 5.7393 5.7411 5.7430 G20 1.5928 1.5863 1.5873 1.5864 G21 2.3455 2.3498 2.3554 2.3602 G22 7.5031 7.4949 7.4897 7.4856 G23 9.2814 9.2948 9.2958 9.3000 G24 −5.4638 −5.4579 −5.4583 −5.4559 G25 −7.1005 −7.1194 −7.1222 −7.1292 G26 −8.5873 −8.5992 −8.6059 −8.6114 G27 −5.1256 −5.1196 −5.1166 −5.1145 G28 3.2658 3.2526 3.2459 3.2399 G29 2.2501 2.2547 2.2610 2.2630 G30 −5.9075 −5.9087 −5.9144 −5.9158 G31 5.0344 5.0264 5.0264 5.0244 G32 −4.4046 −4.4053 −4.3915 −4.3886
Table 2 Maximum, minimum, and differences in G15 and G12 DCB estimates based on different spacings using GRCC (a) and GRCD (b) observations [ns].
1 2.6576 2.6321 0.0255 2 2.6544 2.6300 0.0244 3 2.7129 2.6796 0.0333 4 2.6271 2.5960 0.0311 5 2.6815 2.6415 0.0400 6 2.6331 2.5947 0.0384 7 2.7436 2.7234 0.0202 8 2.6172 2.5813 0.0359 9 2.6517 2.6226 0.0291 10 2.6460 2.6130 0.0330 11 2.5914 2.5675 0.0239 12 2.7338 2.7149 0.0189 13 2.6573 2.6311 0.0262 14 2.6595 2.6303 0.0292 15 2.6239 2.5947 0.0292 16 2.7730 2.7469 0.0261 17 2.5507 2.5257 0.0250 18 2.7313 2.7119 0.0194 19 2.6108 2.5838 0.0270 20 2.5493 2.5269 0.0224 21 2.6050 2.5789 0.0261 22 2.5817 2.5608 0.0209 23 2.5638 2.5429 0.0209 24 2.6731 2.6508 0.0223 25 2.6441 2.6113 0.0328 26 2.7022 2.6726 0.0296 27 2.7357 2.7116 0.0241 28 2.7491 2.7288 0.0203 29 2.6788 2.6558 0.0230 30 2.7242 2.7011 0.0231 1 4.1929 4.1704 0.0225 2 4.3165 4.2822 0.0343 3 4.1862 4.1428 0.0434 4 4.3340 4.2874 0.0466 5 4.2277 4.1950 0.0327 6 4.2023 4.1731 0.0292 7 4.0969 4.0764 0.0205 8 4.1886 4.1454 0.0432 9 4.2370 4.1940 0.0430 10 4.2623 4.2294 0.0329 11 4.2704 4.2374 0.0330 12 4.2513 4.2283 0.0230 13 4.3040 4.2759 0.0281 14 4.1481 4.1161 0.0320 15 4.2978 4.2638 0.0340 16 4.3196 4.2997 0.0199 17 4.2499 4.2316 0.0183 18 4.2684 4.2436 0.0248 19 4.2586 4.2221 0.0365 20 4.2246 4.1879 0.0367 21 4.3187 4.2914 0.0273 22 4.2345 4.2127 0.0218 23 4.2593 4.2333 0.0260 24 4.2821 4.2526 0.0295 25 4.2135 4.1811 0.0324 26 4.2052 4.1790 0.0262 27 4.2133 4.1931 0.0202 28 4.2085 4.1896 0.0189 29 4.1718 4.1506 0.0212 30 4.2738 4.2484 0.0254
Table 3 Mean STD of GPS DCBs adopting different modeling spacings [ns].
LEO Var. STD LEO Var. STD GRCC 2H 0.0624 GRCD 2H 0.0648 4H 0.0621 4H 0.0643 6H 0.0620 6H 0.0641 12H 0.0619 12H 0.0639
Table 4 Mean RMS statistics for GPS DCBs using different modeling spacings [ns].
LEO Var. RMS LEO Var. RMS GRCC 2H 0.2801 GRCD 2H 0.2763 4H 0.2769 4H 0.2732 6H 0.2759 6H 0.2722 12H 0.2748 12H 0.2711
Table 5 Mean values and STD statistics for receiver DCBs with different modeling spacings [ns].
LEO Var. Mean STD GRCC 2H −1.0684 0.0465 4H −1.0202 0.0459 6H −0.9833 0.0456 12H −0.9115 0.0458 GRCD 2H 4.4847 0.0439 4H 4.5365 0.0434 6H 4.5761 0.0450 12H 4.6531 0.0468
Table 6 Mean GPS DCB estimates based on different modeling D&Os using GRCC (a) and GRCD (b) data [ns].
G01 −7.4142 −7.4174 −7.4178 G02 9.1261 9.1298 9.1342 G03 −5.0144 −5.0161 −5.0181 G05 2.9686 2.9782 2.9862 G06 −6.6785 −6.6762 −6.6719 G07 3.3560 3.3536 3.3555 G08 −7.5223 −7.5226 −7.5236 G09 −4.9485 −4.9469 −4.9473 G10 −5.0425 −5.0449 −5.0479 G11 3.7338 3.7332 3.7316 G12 4.1486 4.1480 4.1491 G13 3.4254 3.4277 3.4261 G14 1.9223 1.9205 1.9152 G15 2.6427 2.6467 2.6493 G16 2.7812 2.7844 2.7874 G17 2.9411 2.9406 2.9395 G18 −0.2882 −0.2915 −0.2940 G19 5.6420 5.6405 5.6387 G20 1.4615 1.4640 1.4668 G21 2.2761 2.2791 2.2807 G22 7.2670 7.2648 7.2637 G23 9.1425 9.1434 9.1420 G24 −5.5488 −5.5542 −5.5550 G25 −7.1203 −7.1219 −7.1223 G26 −8.5842 −8.5834 −8.5830 G27 −5.2361 −5.2366 −5.2396 G28 3.1981 3.1953 3.1957 G29 2.2158 2.2135 2.2079 G30 −6.0024 −6.0065 −6.0056 G31 4.9613 4.9588 4.9594 G32 −4.3729 −4.3735 −4.3770 G01 −7.2891 −7.2886 −7.2899 G02 9.3046 9.3094 9.3138 G03 −4.8226 −4.8174 −4.8238 G05 3.0169 3.0263 3.0372 G06 −6.6027 −6.6008 −6.5959 G07 3.4177 3.4194 3.4179 G08 −7.2217 −7.2240 −7.2245 G09 −4.7683 −4.7650 −4.7695 G10 −4.9894 −4.9938 −4.9927 G11 3.8018 3.8041 3.7994 G12 4.2273 4.2284 4.2246 G13 3.4802 3.4804 3.4802 G14 2.0213 2.0114 2.0155 G15 2.6549 2.6580 2.6601 G16 2.9295 2.9338 2.9345 G17 2.9945 2.9941 2.9910 G18 −0.2099 −0.2115 −0.2140 G19 5.7406 5.7393 5.7359 G20 1.5854 1.5863 1.5916 G21 2.3490 2.3498 2.3540 G22 7.4943 7.4949 7.4922 G23 9.2922 9.2948 9.2901 G24 −5.4554 −5.4579 −5.4609 G25 −7.1148 −7.1194 −7.1166 G26 −8.6021 −8.5992 −8.6001 G27 −5.1179 −5.1196 −5.1212 G28 3.2543 3.2526 3.2514 G29 2.2605 2.2547 2.2539 G30 −5.9095 −5.9087 −5.9124 G31 5.0311 5.0264 5.0287 G32 −4.3943 −4.4053 −4.3967
Table 7 Maximum, minimum, and differences in G05 DCB estimates with different D&Os using GRCC (a) and GRCD (b) data [ns].
1 2.9753 2.9694 0.0059 2 2.9370 2.9083 0.0287 3 3.0535 3.0279 0.0256 4 3.0784 3.0584 0.0200 5 3.0444 3.0316 0.0128 6 3.0264 3.0202 0.0062 7 2.9724 2.9534 0.0190 8 2.9780 2.9507 0.0273 9 2.9906 2.9638 0.0268 10 2.9440 2.9432 0.0008 11 2.9463 2.9455 0.0008 12 3.0405 3.0269 0.0136 13 3.0099 2.9776 0.0323 14 3.0311 2.9985 0.0326 15 2.9426 2.9201 0.0225 16 3.0109 3.0031 0.0078 17 2.9305 2.9231 0.0074 18 2.9671 2.9454 0.0217 19 2.9606 2.9328 0.0278 20 2.9821 2.9593 0.0228 21 2.8937 2.8820 0.0117 22 3.0197 3.0148 0.0049 23 2.9796 2.9683 0.0113 24 3.0198 2.9933 0.0265 25 2.9709 2.9403 0.0306 26 3.0022 2.9846 0.0176 27 2.9335 2.9278 0.0057 28 2.9700 2.9640 0.0060 29 2.9186 2.8980 0.0206 30 3.0551 3.0252 0.0299 1 2.9849 2.9719 0.0130 2 2.9826 2.9522 0.0304 3 3.0770 3.0400 0.0370 4 3.0551 3.0259 0.0292 5 3.0523 3.0351 0.0172 6 3.0177 3.0086 0.0091 7 3.0611 3.0390 0.0221 8 3.0289 3.0009 0.0280 9 3.0317 2.9964 0.0353 10 2.9661 2.9488 0.0173 11 3.0090 3.0080 0.0010 12 3.1287 3.1131 0.0156 13 3.0902 3.0569 0.0333 14 3.0515 3.0048 0.0467 15 2.9874 2.9643 0.0231 16 3.0414 3.0352 0.0062 17 2.9714 2.9672 0.0042 18 3.0320 3.0124 0.0196 19 3.0503 3.0195 0.0308 20 3.0253 3.0044 0.0209 21 2.9572 2.9451 0.0121 22 3.0951 3.0901 0.0050 23 3.0627 3.0507 0.0120 24 3.1039 3.0770 0.0269 25 3.0341 3.0054 0.0287 26 3.0661 3.0466 0.0195 27 2.9857 2.9789 0.0068 28 3.0339 3.0263 0.0076 29 3.0092 2.9864 0.0228 30 3.1241 3.0964 0.0277
Table 8 Mean STD of GPS DCBs using different modeling D&Os [ns].
LEO Var. STD LEO Var. STD GRCC 6D&O 0.0620 GRCD 6D&O 0.0643 8D&O 0.0621 8D&O 0.0643 10D&O 0.0623 10D&O 0.0644
Table 9 Mean RMS statistics for GPS DCBs using different modeling D&Os [ns].
LEO Var. RMS LEO Var. RMS GRCC 6D&O 0.2780 GRCD 6D&O 0.2742 8D&O 0.2769 8D&O 0.2732 10D&O 0.2768 10D&O 0.2732
Table 10 Mean values and STD statistics for receiver DCBs with different modeling D&Os [ns].
LEO Var. Mean STD GRCC 6D&O −1.0132 0.0460 8D&O −1.0202 0.0459 10D&O −1.0215 0.0457 GRCD 6D&O 4.5439 0.0449 8D&O 4.5365 0.0434 10D&O 4.5353 0.0441
In this work, M.L. proposed the idea, designed the experiment results, and provided the software; Y.W. collected the experimental data; Y.W. and M.L. analyzed the experiment results and completed the paper. Y.Y., G.W. and H.G. supervised its analysis and edited the manuscript. All authors have read and agreed to the published version of the manuscript.
CODE precise products are available at ftp://ftp.aiub.unibe.ch/ (accessed on 27 February 2023); onboard GPS observation data of the GRACE-FO satellites are available at ftp://rz-vm152.gfz-potsdam.de/ (accessed on 27 February 2023).
The authors declare that this study received funding from China Southern Power Grid Yunnan Power Grid Co., Ltd. (YNKJXM20220033). The funder had the following involvement with the study: Yifan Wang and Mingming Liu analyzed the experiment results and completed the paper, Yunbin Yuan, Guofang Wang, and Hao Geng supervised its analysis and edited the manuscript.
The authors are grateful to CODE for providing the precise products and onboard GPS observation data of GRACE-FO satellites, respectively. Meanwhile, we would like to thank the anonymous reviewers for their valuable comments.
By Yifan Wang; Mingming Liu; Yunbin Yuan; Guofang Wang and Hao Geng
Reported by Author; Author; Author; Author; Author